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Theorem imrestr 25098
Description: Image of an element of transitive class  B under a class restricted by  B. (Contributed by FL, 20-Mar-2011.)
Assertion
Ref Expression
imrestr  |-  ( ( Tr  B  /\  A  e.  B )  ->  (
( C  |`  B )
" A )  =  ( C " A
) )

Proof of Theorem imrestr
StepHypRef Expression
1 trss 4122 . . 3  |-  ( Tr  B  ->  ( A  e.  B  ->  A  C_  B ) )
2 resima2 4988 . . 3  |-  ( A 
C_  B  ->  (
( C  |`  B )
" A )  =  ( C " A
) )
31, 2syl6 29 . 2  |-  ( Tr  B  ->  ( A  e.  B  ->  ( ( C  |`  B ) " A )  =  ( C " A ) ) )
43imp 418 1  |-  ( ( Tr  B  /\  A  e.  B )  ->  (
( C  |`  B )
" A )  =  ( C " A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   Tr wtr 4113    |` cres 4691   "cima 4692
This theorem is referenced by:  imresord  25099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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